Complementary bases in symplectic matrices and a proof that their determinant is one

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1 Linear Algebra and its Applications 419 (2006) Complementary bases in symplectic matrices and a proof that their determinant is one Froilán M. Dopico a,, Charles R. Johnson b a Departamento de Matemáticas, Universidad Carlos III de Madrid, Avda. de la Universidad, 30, Leganés, Madrid, pain b Department of Mathematics, The College of William and Mary, Williamsburg, VA , UA Received 27 July 2004; accepted 13 June 2006 Available online 4 August 2006 ubmitted by V. Mehrmann Abstract New results on the patterns of linearly independent rows and columns among the blocks of a symplectic matrix are presented. These results are combined with the block structure of the inverse of a symplectic matrix, together with some properties of chur complements, to give a new and elementary proof that the determinant of any symplectic matrix is +1. The new proof is valid for any field. Information on the zero patterns compatible with the symplectic structure is also presented Elsevier Inc. All rights reserved. AM classification: 15A15; 15A03; 15A09; 15A57 Keywords: Complementary bases; Determinant; Patterns of linearly independent rows and columns; Patterns of zeros; chur complements; ymplectic The research of F.M. Dopico was partially supported by the Ministerio de Educación y Ciencia of pain through grant BFM , and by the PRICIT Program of the Comunidad de Madrid through IMUMAT Project (Grant -0505/EP/0158). The research of C.R. Johnson was partially carried out during a visit to the Universidad Carlos III de Madrid in November 2003, supported by the Ph.D. Program of Ingeniería Matemática. Corresponding author. addresses: dopico@math.uc3m.es (F.M. Dopico), crjohnso@math.wm.edu (C.R. Johnson) /$ - see front matter ( 2006 Elsevier Inc. All rights reserved. doi: /j.laa

2 1. Introduction F.M. Dopico, C.R. Johnson / Linear Algebra and its Applications 419 (2006) Let I n denote the n-by-n identity matrix and J the 2n-by-2n matrix 0 In J :=. I n 0 J is an orthogonal, skew-symmetric matrix, so that J 1 = J T = J. Definition 1.1. A2n-by-2n matrix with entries in the field K is called symplectic, or J -orthogonal, if T J = J. The group of the symplectic matrices is a relevant class of matrices both from a pure mathematical point of view [9], and from the point of view of applications. For instance, the symplectic matrices play an important role in classical mechanics and Hamiltonian dynamical systems [1,2]. Moreover, the symplectic matrices appear in the linear control theory for discrete-time systems [4,10]. Notice that the symplectic matrices represent the isometries of the skew-symmetric bilinear form defined by J, i.e., x,y =x T Jy, where x and y are 2n-by-1 column vectors. If the field K is C, one can also consider the sesquilinear form x,y =x Jy, where x is the conjugate transpose of x. In this case, the isometries correspond to those matrices such that J = J. These matrices are called conjugate symplectic matrices 1 in the literature [3,7]. We will not deal explicitly with conjugate symplectic matrices, however we will remark in the proper places that some of the presented results remain also valid for these matrices. The following properties are very easily proved from Definition 1.1: the product of two symplectic matrices is also symplectic; if is symplectic then 1 and T are symplectic; and λ K is an eigenvalue of if and only if λ 1 K is an eigenvalue of. Definition 1.1 and the multiplicativity of the determinant imply directly that det =±1. However, the determinant of a symplectic matrix is, actually, always +1, for any field K. If the elements of the field K can be considered complex numbers (for instance if K is the field of real numbers or the field of rational numbers), the property det =+1 is equivalent to the fact that if 1isan eigenvalue of a symplectic matrix then its algebraic multiplicity is even. Also, it is equivalent to the fact that if +1 is an eigenvalue of a symplectic matrix then its algebraic multiplicity is also even. These two properties can be seen as follows: notice that = J 1 T J. This means that if λ/= λ 1 is an eigenvalue of, then λ 1 is also an eigenvalue of with the same algebraic multiplicity as λ. Taking into account that the dimension of is even, the sum of the algebraic multiplicities of +1 and 1 (if some of them is an eigenvalue) is even. Moreover det() = ( 1) a, where a is the algebraic multiplicity of 1. It has long been known that the determinant of any symplectic matrix is +1, but no proof seems entirely elementary. This has been recognized by several authors. Let us quote, for instance, the following paragraph from [9]: It is somewhat nontrivial to prove that the determinant itself is +1, and we will accomplish this by expressing the condition for a matrix to be symplectic in terms of a differential form Another proof may be found in Arnold [2]. 1 The matrices satisfying J = J and having complex entries are also frequently called in the literature complex symplectic matrices [6]. We do not use this name, because it may produce confusion when compared with Definition 1.1 in the case K = C.

3 774 F.M. Dopico, C.R. Johnson / Linear Algebra and its Applications 419 (2006) Or from the very recent work [8]: The aim of this essay is to shine some light on this unexpected result (det() =+1) from various angles, hoping to demystify it to some degree. Our purpose here is to give a straightforward matrix theoretical proof that det =+1 when is symplectic, but in the process, we give new information about the patterns of linearly independent rows and columns among the blocks of a symplectic matrix. We refer to these relationships as the complementary bases theorem. The essential ingredients of our work are very simple: basic properties of chur complements and the special block structure of the inverse of a symplectic matrix. One essential difference between the proof we present for det =+1 and those presented in [2,8,9], is that the new proof looks at the elements of the matrix, while the others pay attention to factorization properties of the symplectic group, or to other more abstract properties of symplectic matrices. 2. Preliminary results In this section, we summarize some well-known basic results that will be used later on. Moreover, an extremely elementary proof of the determinant of certain symplectic matrices being one is presented. Unfortunately, this proof is not valid for an arbitrary symplectic matrix, but it is one of the fundamental ideas in our work. Recall [5, ection 0.7.3, p. 18] that if an m-by-m invertible matrix A is partitioned as A11 A A =, A 21 A 22 in which A 11 is nonsingular, and if A 1 B11 B = B 21 B 22 is partitioned conformally, we then have that B 22 is nonsingular, and B 1 22 = A 22 A 21 A 1 11 A, i.e., B 1 22 is the chur complement of A 11 in A. Therefore deta = deta 11 det[a 22 A 21 A 1 11 A ] = det A 11. (1) det B 22 On the other hand, any 2n-by-2n matrix is naturally thought of in partitioned form 11 =, (2) in which 11 is n-by-n and we shall fix and refer to this partition throughout. A matrix so partitioned is symplectic if and only if [ T 1 = 22 T ] 21 T 11 T. (3) This is easily proven: Definition 1.1 implies that is symplectic if and only if 1 = J 1 T J = J T J. Then some trivial block manipulations lead to the result above.

4 F.M. Dopico, C.R. Johnson / Linear Algebra and its Applications 419 (2006) Applying the determinantal formula (1), we quickly obtain that if a symplectic matrix has the block 11 nonsingular then det = det 11 det11 T = det 11 = 1. det 11 The fact that det =+1 may also be easily obtained whenever at least one of the four blocks in (2) is nonsingular: simply perform some row and/or column interchanges in (2) tomove the nonsingular block to the (1, 1)-position, and follow the track of these interchanges in the block expression of the inverse (3). Finally, (1) is applied to the resulting matrix. Let us explain with more detail this process in the specific case of being nonsingular and 11 being singular. Let P be the permutation matrix that interchanges in the jth column of 11 with the jth column of for all j {1,...,n}. Thus P has as its (1, 1)-block, and (P ) 1 = P T 1 has T as its (2, 2)-block, because P T interchanges the corresponding rows of 1. Eq. (1) implies that det(p ) = det /det( ) = ( 1) n. Our claim follows from the fact that det(p ) = ( 1) n det. The case 21 being nonsingular can be dealt with in the same way, interchanging rows of instead of columns. Finally, if one wants to move 22 to the (1, 1)-block position, both rows and columns have to be interchanged. This is the simplest case because no sign changes appear. Unfortunately, the following example shows that there exist symplectic matrices having the four blocks singular: In the next section, the notion of complementary bases is introduced. This idea allows us to relax the restriction of at least one of the blocks being nonsingular in the above proof, making it entirely general. To finish this section, let us comment that for conjugate symplectic matrices, i.e., those fulfilling J = J, the inverse of has the structure appearing in (3), but replacing transpose by conjugate transpose matrices. Thus, in this case, (1) leads to det = det 11 det 11, which is not necessarily +1, but necessarily on the unit circle. 3. The complementary bases theorem The main new result in this work is easily described in partitioned form in the next theorem. In this theorem, α denotes the cardinality of a set α. Moreover the binary variables p and q can take as values 1 or 2, and p and q denote, respectively, the complementary variables of p and q. Theorem 3.1. Let be a 2n-by-2n symplectic matrix partitioned as in (2). uppose that rank ( pq ) = k, p, q {1, 2}, and that the rows (columns) of pq indexed( by α, ) α {1,...,n} and α =k, are linearly independent. Then the rows (columns) of p q pq indexed by α, the complement of α, together with the rows (columns) αof pq constitute a basis of K n, i.e., they constitute a nonsingular n-by-n matrix. The fact that certain rows (columns) from one block, together with complementary rows (columns) from another constitute a basis is the motivation for the name we have chosen. In fact,

5 776 F.M. Dopico, C.R. Johnson / Linear Algebra and its Applications 419 (2006) Theorem 3.1 gathers eight different results: one for rows and the other for columns for each of the four blocks. As an specific illustration of the complementary bases result, let us consider a symplectic matrix with rank 11 = 2, 11 =[a 1,a 2,a 3,a 4,a 5 ], and =[b 1,b 2,b 3,b 4,b 5 ], where a i and b j denote columns. Assume that {a 1,a 4 } is a linearly independent set of columns, then the complementary bases result states that the 5 5 matrix [a 1,b 2,b 3,a 4,b 5 ] is nonsingular. It should be noticed that is nonsingular. Therefore, the n-by-2n matrix [ 11 ] has rank equal to n, and also has n linearly independent columns. The complementary bases theorem specifies, for symplectic matrices, a very particular set of n independent columns by using complementarity. This phenomenon need not occur within a general 2n-by-2n partitioned nonsingular matrix, as the reader can easily check by devising some examples. Proof of Theorem 3.1. We only need to prove the theorem in the column case p = 1 and q = 1. This result applied on the symplectic matrix J proves the column case p = 1 and q = 2. Once these two cases are proved, the other six cases follow from (2) and (3), and the fact that 1, T, and T are, all of them, symplectic matrices. We begin by selecting an n-by-n permutation matrix P to move the maximal linearly independent set of columns of 11 indexed by α to the first k positions. This is accomplished by multiplying on the right by the symplectic matrix P P to get the symplectic matrix : [ P 0 := = 11 ] 0 P (4) Let Y be the n-by-n nonsingular matrix such that Y 11 is a row reduced echelon matrix, and let us multiply on the left by the symplectic matrix Y Y T to get the symplectic matrix : [ Y 0 := 0 Y T 11 ] [ 11 ] = 21 22, where 11 = Ik Z and 0 0 = X11 X (5) X 21 X 22 are partitioned conformally. Obviously, the multiplication above does not change the relationships of linear independence among the columns of [ 11 ], therefore to prove the complementary bases theorem is equivalent to prove that the (n k)-by-(n k) matrix X 22 is nonsingular. Bearing in mind that is symplectic, its inverse has the block structure appearing in (3), then 0 = ( 1 ) = 11 T + T 11, and substituting (5), we get [ X11 X11 T + X Z T ZX T (X21 T + ZXT 22 ) ] X 21 + X 22 Z T = 0. 0 This implies X 21 = X 22 Z T. Any subset of rows of is linearly independent because is nonsingular. In particular all the rows of the submatrix [0 0X 21 X 22 ] are linearly independent, thus n k = rank[x 21 X 22 ]=rankx 22 [ Z T I]=rankX 22.

6 F.M. Dopico, C.R. Johnson / Linear Algebra and its Applications 419 (2006) An immediate corollary of the complementary bases theorem gives information on the patterns of zeros compatible with the symplectic structure: Corollary 3.2. Let be a 2n-by-2n symplectic matrix partitioned as in (2). uppose that the rows (columns) ( of) pq,p,q {1, 2}, indexed by β, β {1,...,n}, are zero. Then the rows (columns) of p q pq indexed by β are linearly independent. Proof. Every set of indices α considered in Theorem 3.1 is included in β. Then β is included in α. It should be noticed that Theorem 3.1 and Corollary 3.2 are valid for conjugate symplectic matrices. The statement of Theorem 3.1 suggests the following natural question: may the complementary ( ) bases theorem be extended to rows (columns) of pq and rows (columns) of pq p q? Notice, that this would mean eight additional results. However, the following example shows that this extension is not true. Let us consider the symplectic matrix The rank of the (1, 1)-block is obviously 1 and its second row constitutes a maximal linearly independent set of rows. However, the second row of the (1, 1)-block together with the first row of the (1, 2)-block do not constitute a nonsingular matrix. 4. A proof that the determinant of a symplectic matrix is one In this last section we prove, as a corollary of Theorem 3.1, the classical result which has motivated our work: Theorem 4.1. Let be a 2n-by-2n symplectic matrix with entries in any field K. Then det =+1. Proof. Let us consider again the partition (2) and assume that rank( 11 ) = k. Let us select as in (4) ann-by-n permutation matrix P to move a maximal linearly independent set of columns of 11 to the first k positions. Notice that = (P P) is symplectic and that det = det, because det(p P) = (detp) 2 = 1. From now on, we will focus on the matrix. Let us consider the following partition of the two upper blocks of : [ 11 ] =[E1 E 2 F 1 F 2 ], where E 1 and F 1 are n-by-k matrices, and E 2 and F 2 are n-by-(n k) matrices. Then, Theorem 3.1 implies that the n-by-n matrix [E 1 F 2 ] is nonsingular. Let Π j be the 2n-by-2n matrix that multiply by 1 the column j of 11, and then interchanges this column with the column j of. Therefore, Π j has determinant +1 and has the following block structure [ Q1 Q Π j = 2 Q 2 Q 1 ],

7 778 F.M. Dopico, C.R. Johnson / Linear Algebra and its Applications 419 (2006) where Q 1 is an n-by-n diagonal matrix with ones on the diagonal, except for a zero in the (j, j) position; and Q 2 has all the entries zero, except +1 in the (j, j) position. It can be trivially checked that Π j is symplectic. Then E1 F Π k+1 Π n = 2 is also symplectic, and, taking into account the structure of the inverse of a symplectic matrix (3), ( Π k+1 Π n ) 1 = E1 T. F T 2 Now, the determinantal formula (1) is applied to the matrix Π k+1 Π n : det Π k+1 Π n = det[e 1F 2 = det[e 1F 2 ] E T det 1 det[e 1 F 2 ] T = 1. F T 2 Finally, det Π k+1 Π n = det = det. Then det = 1. Acknowledgments The authors thank Prof. Alberto Ibort for many illuminating discussions on classical and not so classical proofs of the result det =+1. We also thank David Martínez for his careful explanation of a fully algebraic proof, without differential forms, using exterior algebra. Prof. Volker Mehrmann informed us of Ref. [8] after this work was presented in the GAMM Workshop on Applied and Numerical Linear Algebra (Hagen, July 2004), and we thank him for many enthusiastic and interesting discussions on the topics of Hamiltonian and symplectic matrices during the last two years. Finally, we sincerely thank the referees for their careful reading of the manuscript and for many helpful suggestions that have improved significantly the paper. References [1] R. Abraham, J. Marsden, Foundations of Mechanics, second ed., Addison-Wesley, Reading, [2] V.I. Arnold, Mathematical Methods in Classical Mechanics, pringer-verlag, Berlin, [3] A. Bunse-Gerstner, R. Byers, V. Mehrmann, A chart of numerical methods for structured eigenvalue problems, IAM J. Matrix Anal. Appl. 13 (2) (1992) [4] H. Fassbender, ymplectic Methods for the ymplectic Eigenproblem, Kluwer, New York, [5] R.A. Horn, C.R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, [6] W.-W. Lin, V. Mehrmann, H. Xu, Canonical forms for Hamiltonian and symplectic matrices and pencils, Linear Algebra Appl (1999) [7] D.. Mackey, N. Mackey, F. Tisseur, tructured tools for structured matrices, Electron. J. Linear Algebra 10 (2003) [8] D.. Mackey, N. Mackey, On the determinant of symplectic matrices, Numerical Analysis Report No. 422, Manchester Centre for Computational Mathematics, Manchester, England, [9] D. McDuff, D. alamon, Introduction to ymplectic Topology, Clarendon Press, Oxford, [10] V. Mehrmann, The Autonomous Linear Quadratic Control Problem Theory and Numerical olution, Lecture Notes in Control and Information ciences, vol. 163, pringer-verlag, Berlin, 1991.